How to solve a recurrence equation with non-constant coefficients? The equation is
$$
120(3k+1)(18k^3-21k-2)(k+2)a_{3k-6}=120(54k^4-117k^2+4)(k-1)a_{3k-5}+(k-1)^2(3k^2-2)^2k^3(k+2)(k+1)(6k^2+6k-1).
$$
Here $a_0=7, a_1=42, a_2=189, a_3=708, a_4=2121$. This equation has non-constant coefficient. Are there some general method to solve this equation?

Thank you very much.

Edit: the other two equations are
$$
(3k+2)(18k^3+54k^2+33k-1)a_{3k-5} = (3k+1)(18k^3-21k-2)a_{3k-4}+(1/120)k^2(3k^2+6k+1)(k+1)^2(3k^2-2)(k-1)(k+2)(6k^2+6k-1).
$$

I think one method can be as follows: assume that $a_{3k}=\sum_{i=1}^{8} c_ik^i, a_{3k+1}=\sum_{i=1}^{8} d_ik^i, a_{3k+2}=\sum_{i=1}^{8} e_ik^i$ and compute $a_1, \ldots, a_{24}$. Then plugin $k=1, \ldots, 24$ and solve $c_1, \ldots, c_8, d_1, \ldots, d_8, e_1, \ldots, e_8$. But it is complicated. Are there some simpler method?